An application of Groebner bases to the classification of nonlinear circuits

One of the application of Groebner base to the classification of nonlinear circuits is proposed. The circuits are excited by a sinusoidal electric voltage or current sources with an angular frequency. In an evaluation of electric circuits the first thing we have to do is to estimate the amplitude characteristic which is defined as the polynomial equation of two variables. These variables are the amplitude of the sources and that of harmonic component of the current or voltage in the circuit elements. The estimation tends to be performed on individual nonlinear circuit. If we are able to classify the nonlinear circuits into several groups based on the amplitude characteristcs, we will have no need to investigate each nonlinear circuit. In order to have a Groebner base we first formulate the circuit equations as the sets of the nonlinear ordinary differential equation of several circuits with specific number of reactive elements such as capacitors and/or inductors. Applying a harmonic balance method to the circuit equations, we derive a simultaneous polynomial equations, which represents the periodic equilibrium state of the circuits. By adding the amplitude characteristic to the simultaneous polynomial equations we have another sets of polynomial equations in which each variable is the amplitudes of sine and cosine functions. Here we try to find a Groebner base for each set of polynomial equations using the lexicographical order. As the result, the last element of the Groebner base which makes the classification possible gives the amplitude characteristic of each circuit. As an example, a Groebner base is shown to classify the nonlinear circuits which are composed by one voltage source (sinusoidal function), one resistor, one capacitor and one nonlinear inductor of which characteristics is given by the polynomial function of the fluxinterlinkages. From these four circuits elements we can compose eight nonlinear circuits. Usually the different circuits are considered to have different amplitude characteristics. However, we have three types of the amplitude characteristics from the different circuits. By use of a Groebner base these eight circuits are classified into three categories from the point of view of the amplitude characteristics of fundamental harmonic oscillations.